#10196173

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#51125	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4119 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,7)(3,4,8,12)(5,13,15,9)(10,14,16,11)(17,18,23,25)(19,28,22,24)(20,29,30,26)(21,27,32,31)(33,34,38,41)(35,39,44,47)(36,37,45,40)(42,43,48,46);(1,3)(2,4)(5,10)(6,8)(7,12)(9,11)(13,14)(15,16)(17,19,23,22)(18,24,25,28)(20,27,30,31)(21,26,32,29)(33,35,38,44)(34,39,41,47)(36,42,45,48)(37,43,40,46);(2,7)(3,8)(9,13)(10,16)(18,25)(20,30)(24,28)(27,31)(33,36,38,45)(34,40,41,37)(35,42,44,48)(39,46,47,43);(1,3)(2,4)(5,10)(6,8)(7,12)(9,11)(13,14)(15,16)(17,20,23,30)(18,26,25,29)(19,27,22,31)(21,24,32,28)(33,36)(34,37)(35,42)(38,45)(39,43)(40,41)(44,48)(46,47);(1,4)(2,8)(3,7)(5,14)(6,12)(9,10)(11,15)(13,16)(17,21,23,32)(18,27,25,31)(19,29,22,26)(20,28,30,24)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,5,6,15)(2,9,7,13)(3,10,8,16)(4,11,12,14)(17,22,23,19)(18,28,25,24)(20,31,30,27)(21,29,32,26)(33,37)(34,36)(35,43)(38,40)(39,42)(41,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10103369);